Question: Solve for $z$, $ -\dfrac{7}{15z + 12} = -\dfrac{3z - 6}{5z + 4} + \dfrac{10}{5z + 4} $
First we need to find a common denominator for all the expressions. This means finding the least common multiple of $15z + 12$ $5z + 4$ and $5z + 4$ The common denominator is $15z + 12$ The denominator of the first term is already $15z + 12$ , so we don't need to change it. To get $15z + 12$ in the denominator of the second term, multiply it by $\frac{3}{3}$ $ -\dfrac{3z - 6}{5z + 4} \times \dfrac{3}{3} = -\dfrac{9z - 18}{15z + 12} $ To get $15z + 12$ in the denominator of the third term, multiply it by $\frac{3}{3}$ $ \dfrac{10}{5z + 4} \times \dfrac{3}{3} = \dfrac{30}{15z + 12} $ This give us: $ -\dfrac{7}{15z + 12} = -\dfrac{9z - 18}{15z + 12} + \dfrac{30}{15z + 12} $ If we multiply both sides of the equation by $15z + 12$ , we get: $ -7 = -9z + 18 + 30$ $ -7 = -9z + 48$ $ -55 = -9z $ $ z = \dfrac{55}{9}$